
to the ideal GKP states with noise of variance \(\frac{\delta }{2}\)  [1]}, [2]} in both quadratures, with \(\delta =\Delta ^2\)  from eq:finiteEGKP.
While this noise model does not capture the peak-damping envelope of eq:finiteEGKP, it captures the finite width added to each delta-function in phase space.
In our case, we find that the same noise model framework can be used to model the replacement of \(\mathinner {|{+}\rangle }_{\rm {gkp}}\)  states with \(p\) -squeezed states, setting \(\epsilon = \Delta ^{2} = \delta \)  from eq:CVCS.
In particular, we notice that adding Gaussian noise of variance \(\delta /2\)  (\(\frac{1}{2\delta }\) ) in \(p\)  (\(q\) ) quadrature makes the \(\mathinner {|{+}\rangle }_{\rm {gkp}}\)  mimic the Wigner function of a mixture of \(p\) -squeezed states.
In the context of eq:noise, the noise matrices for GKP and \(p\) -squeezed states are given by:
\(\mathbf {Y}_{\rm gkp} = \frac{1}{2}\begin{pmatrix} \delta & 0 \\ 0 & \delta \end{pmatrix}, ~~ \mathbf {Y}_{\rm p} = \frac{1}{2}\begin{pmatrix} \delta ^{-1} & 0 \\ 0 & \delta \end{pmatrix}.\) 
